Calculus
Derivatives, integrals and Taylor expansions in one and several variables.
5 lessons open · 9 planned
Pages you can open
What a derivative measures when you zoom in
What does the number f′(a) tell you about the graph of f?
Why the chain rule multiplies
When x goes into g and the result goes into f, how fast does f(g(x)) change?
What the fundamental theorem of calculus connects
Slopes and areas look like two separate problems. What links the slope of a graph to the area under it?
Where e comes from
A bank pays interest at a nominal annual rate of 100%. If it compounds more and more often, monthly, then daily, then every second, how much does $1 grow to in one year?
What degree does a Taylor polynomial need?
A polynomial needs only additions and multiplications. What degree does it take for a polynomial to stay within 0.005 of ex for every x from −1 to 1, and is there always a degree that does the job?
Planned
In learning order. These are not pages yet.
- Partial derivatives and the tangent planeIntro
- Which Taylor terms vanish as Δx shrinksIntro
- When an infinite area is finite: improper integralsIntro
- Why the gradient points uphillIntro
- The multivariable chain rule as a sum over pathsIntro
- Multivariable Taylor expansion: what the Hessian addsIntro
- Change of variables and the Jacobian determinantIntro
- Differentiating under the integral signAdvanced undergrad
- Why the Gaussian integral equals √πIntro
How it connects
Builds on Linear algebra
Leads to Differential equations Probability Numerical methods Geometry Bonds and interest rates Real analysis Optimization Complex analysis Statistical inference Stochastic calculus and SDEs Corporate finance Derivatives Market risk
Subjects marked “connection” are linked by a related idea, not by a prerequisite.